Concentration bulk species

Fig. 5. Concentration polarization = concentration at membrane wall, Cj, = bulk concentration, Cj,. = bulk concentration of species i, J = flux, and...

The situation illustrated in Figure 4 allows both species to coexist. Either of the two sets of curves can be considered the oxidized species the other is the reduced species. The choice depends on whether oxidation or reduction is occurring at the surface. Assume the upper curve is the reduced species and the lower curve is its oxidized form. An appHed voltage has maintained fixed surface concentrations for some period of time including and The concentration profile of the oxidized species decreases at the electrode surface (0 distance) as it is being reduced. Electrolysis therefore results in an increase in the concentration of reduced species at the surface. The concentration profiles approach bulk values far from the surface of the electrode because electrolysis for short times at small electrodes cannot significantly affect the concentrations of species in large volumes of solution. 

For catalytic reactions carried out in the presence of a heterogeneous catalyst, the observed reaction rate could be determined by the rate of mass transfer from the bulk of the reaction mixture and the outer surface of the catalyst particles or the rate of diffusion of reactants within the catalyst pores. Consider a simple first order reaction its rate must be related to the concentration of species S at the outer surface of the catalyst as follows ... 

In this chapter I explained how isotope ratios may be calculated from equations that are closely related, but not identical, to the equations for the bulk species. Extra terms arise in the isotope equations because isotopic composition is most conveniently expressed in terms of ratios of concentrations. I illustrated the use of these equations in a calculation of the carbon isotopic composition of atmosphere, surface ocean, and deep ocean and in the response of isotope ratios to the combustion of fossil fuels. As an alternative application, I simulated the response of the carbon system in an evaporating lagoon to seasonal changes in biological productivity, temperature, and evaporation rate. With a simulation like the one presented here it is quite easy to explore the effects of various perturbations. Although not done here, it would be easy also to examine the sensitivity of the results to such parameters as water depth and salinity. 

An example of different domains in a copper corrosion problem is shown schematically in Fig. 11.2. Four of the domains domains are volumetric. That is, they are three-dimensional, so concentrations of species within these domains might have units of mol/m3, for example. The volumetric domains shown correspond to the gas (G), bulk copper that is being corroded (B), an aqueous layer (A), and a layer in which corrosion products have formed (C). The list of species that can exist in one domain may be (and surely is) different from the species present in another domain. Chemical reaction rates-of-progress within a volumetric domain have units like mol/m3-s. 

The interface between two volumetric domains is designated a surface domain, and its dimensionality is one less than a volumetric domain. Concentrations of species in a surface domain have dimensions of mol/m2, for example. The four types of surface domains shown in Fig. 11.2 are A-G, the interface between the aqueous domain and the gas A-C, the interface between the aqueous domain and the corrosion-product layer C-G, the interface between the corrosion layer and the gas and C-B, the interface between the corrosion layer and the bulk copper layer. Chemical reactions of species residing in one volumetric domain with species in another volumetric domain have to occur at an interface, namely a surface... 

It will be assumed in this section that the mass transport is much more rapid than the redox kinetics, such that the activities or concentrations of species O and R at the electrode-solution interface can be considered as identical to their bulk values (i.e., a = a °l and c = c 1 with i = O, R). The influence of the mass transport on the current-potential response is treated in Sect. 1.8. 

Mass transport gives rise to the appearance of concentration profiles of an electroactive species O like those shown in Fig. 1.20, obtained for the application of a constant potential to a macroelectrode. From this figure it can be seen that there is a region adjacent to the electrode surface where the concentration of species O is different from its bulk value, Cq, and, therefore, mass transport takes place. In the following discussion, diffusion will be the only transport mode considered. The thickness of this diffusion layer, <5real, can be accurately calculated from the concentration profile as the distance from the electrode surface to a point in solution at which the following condition holds ... 

The criterion discussed above is based on the dependence of the surface concentration of the oxidized species with the reversibility degree of the electrode process. So, for a totally irreversible process, the rate of depletion of the surface concentration Cq is much smaller than the mass transport rate process, and therefore, at the formal potential its value should be coincident with the bulk concentration (co(2,°)/coi — l)- In contrast- for reversible electrode reactions, cb(x°)/co = 0.5 (see Eq. (2.20) of Sect. 2.2 for = 0 and y = 1). In order to verify this behavior, the variation of the surface concentration of species O at the formal potential calculated as a function of has been plotted in Fig. 3.5b. From this figure, it can be deduced that at the irreversible limit (i.e., = 0.17),... 

This different behavior can be explained by considering that for a CE mechanism (the reasoning is similar for an EC one), C species is required by the chemical reaction whose equilibrium is distorted in the reaction layer (whose thickness in the simplified dkss treatment is <5r = jDj(k + 2)) and by the electrochemical reaction, which is limited by the diffusion layer (of thickness 8 = yfnDt). For a catalytic mechanism, C species is also required for both the chemical and the electrochemical reactions, but this last stage gives the same species B, which is demanded by the chemical reaction such that only in the reaction layer do the concentrations of species B and C take values significantly different from those of the bulk of the solution. In summary, the catalytic mechanism can reach a true steady-state current-potential response under planar diffusion because its perturbed zone is restricted to the reaction layer <5r, which is independent of time, whereas the distortion of CE (or EC) mechanism is extended until the diffusion layer 8, which depends on time, and a stationary current-potential response will not be reached under these conditions. 

We consider first the situation in which the chemical reaction is considered as irreversible. It is assumed that the surface concentration of species Csoi remains constant and equal to their bulk value, c c (in such a way that the process can be considered as a pseudo-first order process),... 

Moreover, fG(r, qG) is a function which depends on the electrode geometry (see Table 2.3 for several common situations), c is the bulk concentration of species i, v is the scan rate (= AE/r), and Eeq is the equilibrium potential given by Nemst equation. The superindex or subindex G refers to the electrode geometry considered, and qG to the characteristic dimension of the electrode considered. 

The concentrations of species at the interface depend on the mass transport of these species from bulk solution, often described by the mass transfer coefficient /cd. A reversible reaction corresponds to the case where the kinetics of the electrode reaction is much faster than the transport. The kinetics is expressed by a standard rate constant, kQ, which is the rate constant when E = °. So the criterion for a reversible reaction is... 

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